Electron Probability Distribution for the Hydrogen Atom

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The electron in a hydrogen atom is described by the Schrödinger equation. The time-independent Schrödinger equation in spherical polar coordinates can be solved by separation of variables in the form
. The radial and angular components are Laguerre and Legendre functions, thus
and
, respectively. Here,
is the first Bohr radius, and
are the integers in the ranges
(principal quantum number),
(angular momentum quantum number), and
(magnetic quantum number). The probability of finding an electron at a specific location is given by
, where
is the normalization constant, such that
.
Contributed by: Y. Shibuya (April 2014)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The electron is confined to a space of the order of a few angstroms for . For
, the space expands beyond the display limit (
in radius). The patterns for positive and negative
values are the same since only the real part of
is shown.
Snapshot 1: distribution for ,
Snapshot 2: distribution for ,
Snapshot 3: distribution for ,
Reference
[1] J. C. Slater, Modern Physics, New York: McGraw-Hill, 1955.
Permanent Citation